15 research outputs found
An exponentially averaged Vasyunin formula
We prove a Vasyunin-type formula for an autocorrelation function arising from
a Nyman-Beurling criterion generalized to a probabilistic framework. This
formula can also be seen as a reciprocity formula for cotangent sums, related
to the ones proven in [BC13], [ABB17].Comment: This paper has been written from results already stated in a previous
version of another paper in 2018, but has been now submitted separately.
arXiv admin note: text overlap with arXiv:1805.0673
On the distribution of a cotangent sum
Maier and Rassias computed the moments and proved a distribution result for
the cotangent sum on
average over , as . We give a
simple argument that recovers their results (with stronger error terms) and
extends them to the full range . Moreover, we give a density result
for and answer a question posed by Maier and Rassias on the growth of the
moments of .Comment: 10 pages, 2 figure
Period functions and cotangent sums
We investigate the period function of \sum_{n=1}^\infty\sigma_a(n)\e{nz},
showing it can be analytically continued to and studying its
Taylor series. We use these results to give a simple proof of the Voronoi
formula and to prove an exact formula for the second moments of the Riemann
zeta function. Moreover, we introduce a family of cotangent sums, functions
defined over the rationals, that generalize the Dedekind sum and share with it
the property of satisfying a reciprocity formula. In particular, we find a
reciprocity formula for the Vasyunin sum.Comment: 32 pages, 5 figures, revised version. To appear in Algebra & Number
Theor